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How to calculate the minimum distance between two polygons

Question 1

Do you know how to calculate the minimum distance between two polygons if
they don't overlap?

Well, this is rather complex. I start with the basics (don't know if you
already know). I use UPPER letters for vectors and lower letters for numbers.
Well, and sorry that I repeat some things, I wrote the text and realized I
forgot something you have to know for a later step, so I went back and added
it, I didn't write this text line by line. If something isn't clear to you, I
recommend to make some small drawings, I had to do many, too.

A plane is defined by (X - P) * N = 0 where P is a
vector to any point in your plane and N is the normal vector of
your plane. Sometimes another definition is used, which is easier to gain if
you have the corners of a polygon: X = P + a * A + b * B
(a, b are any real numbers). If you know 3 points
X1, X2, X3 of the plane (3 corners of the
polygon), you can get A and B by
A = X2 - X1 (subtract each component of the vector from the same
component of the other vector) and B = X3 - X1 (and you can use
P = X1).

Unfortunately this definition is not good to calculate distances, so you have
to get N out of A and B. A * N
must be 0 and B * N must be 0 (which
means a1 * n1 + a2 * n2 + a3 * n3 = 0 and
b1 * n1 + b2 * n2 + b3 * n3 = 0). Sorry I cannot remember how to
do this, but you have 2 equations with three unknown variables, so you can
choose one of them as you want (just be careful with 0 and not
0), the only difference is that the resulting N
differs in its length.

A line is defined by X = P + v * V where P is any one
point of your line and V is the line's direction (like A
and B of the plane). Again if you know two points X1 and
X2 of your line you get V by V = X2 - X1
(and you can use P = X1).

Use following formula to get the distance between any one point X
and a plane: dist = 1 / n * (X - P) * N where X is a
vector to the point you want to examine and n is the length of
N (you don't need 1/n if N has already
length of 1). If X is (x1, x2, x3) this is

Now the distance between two polygons isn't that simple, because there are
many different cases (and a polygon is more than just a simple plane, even if
it's size is smaller).

What you also need is to calculate the distance between two lines. At first
you need a plane, that is parallel to both lines and includes one of the
lines:

P(plane) = P(line1)
A(plane) = V(line1)
B(plane) = V(line2)

where A and B are two vectors in the plane
(N * A = 0 and N * B = 0), V is a vector in your line
(for polygons, you can take V = X2 - X1 where X2 and
X1 are two corners). Now calculate the distance between this plane
and any one point of line2 using the formula above (any point
because ALL points have the same distance to a plane that's parallel to the
line - nice trick, isn't it?

The last thing we need is not only the minimum distance between two lines, but
the points of the lines, that have minimum distance. You can do this (for the
point of line1 M1 with minimum distance to line2) by
calculating a plane again with

The second plane includes line2 and the point of line1 with the
minimum distance to line2. To get this point of minimum distance,
set P(line1) + v * V(line1) = P(secondplane) + a * A(secondplane) + b * B(secondplane). Solve this, you should get the v and when you set this
v into X = P + v * V of line1 you have the
point X (=M1) of minimum distance.

The bad news: To get M2, you have to repeat this for
line2. Another way would be to take the distance between the lines
(I call it d) and do following:
M2 = M1 + d * 1 / n * N(plane) (or
M2 = M1 - d * 1 / n * N(plane), depends on the direction of
N). The distance between two points X1 and X2
equals the length of the vector X2 - X1.

Okay, these were the basics. Now the different cases, you have to cope with:

1) The planes of both polygons are parallel
(N1 = x * N2):

Transform both polygons the following way: X' = X - P for each
corner of the polygon (where P is any point in your plane). The new
polygons should now be in the same plane. Test whether both polygons overlap
(is not as simple as it sounds, to be honest I don't know how to do that).

1a) They overlap:

The minimum distance is the distance of the two planes (take any one point of
one plane and use the formula above to get the distance to the other plane).

1b) They do not overlap:

Use 2) to calculate the minimum distance.

2) The planes are not parallel (or case 1b):

Calculate the minimum distance of one line of one polygon and one line of the
other polygon. Calculate the points of minimum distance from the lines. The
edges of the polygons do not have infinite length (the lines do have), so
check whether the points of minimum distance are within the polygons (I'd
better say: within the edges of the poygons).

2a) Both points are within the polygons:

Store the minimum distance from the lines.

2b) One point or both points are not within the polygon:

Take the corner(s) of the polygon(s) within the line(s) you checked next to
the point(s) of minimum distance. Calculate the distance between these points
and store it. Now repeat this for each pair of lines (if you have 2 triangles
you get 9 combinations (3 times 3). When you are ready compare all the minimum
distances and take the smallest one.

Okay, this is quite much to do (realtime? difficult. perhaps if you don't have
many polygons) and there are several problems (a vector
(x1, x2, x3) and a vector (x1, x2, 0) may have to be
treated different, for example when you try to get N out of
A and B). If you really need the minimum distance, try
it, but perhaps you find an easier way, that is not that exact (take the
distance between the center of each polygon would be least exact, but very
much easier).

I want to add, that I don't know of other solutions, perhaps there are better
ones, and that I don't know if everything I told you is right, I haven't
tested it, everything is just theoretically.